Contents

Etymology

Our modern word "sine", is derived from the Latin word sinus, which means "bay" or "fold", from a mistranslation (via Arabic) of the Sanskrit word jiva, alternatively called jya.[1]Aryabhata used the term ardha-jiva ("half-chord"), which was shortened to jiva and then transliterated by the Arabs as jiba (جب). European translators like Robert of Chester and Gherardo of Cremona in 12th-century Toledo confused jiba for jaib (جب), meaning "bay", probably because jiba (جب) and jaib (جب) are written the same in the Arabic script (this writing system uses accents instead of vowels and in some formats the accents are not written to ease writing, so if the readers are not familiar with the language they might be confused between words with the same letters but different phonetics). The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae.[2] These roughly translate to "first small parts" and "second small parts".

Early developments

Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.[3] They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.[4]

The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".[5]

Ancient Egyptian mathematics

The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.[6] The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (circa 1680-1620 BC), contains the following problem related to trigonometry:[6]

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[6]

Ancient Indian mathematics

The earliest use of sine appears in the Sulba Sutras written in ancient India from the 8th century BC to the 6th century BC, which correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle), though they had not yet developed the notion of a sine in a general sense.[8]

Hellenistic mathematics

Ancient Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of the bisected angle, that is, , and consequently the sine function is also known as the "half chord". Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.[9]

Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas.[5] For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosine for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the law of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.[5] To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the well known theorem that, in modern notation, whenever , among other theorems.[10]

Asia Minor

An early trigonometric table was apparently said to have been compiled by Hipparchus of Nicaea (180 - 125 BC).[11] Hipparchus apparently tabulated the corresponding values of arc and chord for a series of angles.[1][11]

Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 B.C.), since he measured an angle in terms of a fraction of a quadrant.[10] It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.[12] In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts.[2] It is due to the Babylonian sexagesimal number system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.[2]

Hellenistic Egypt

In Roman Egypt, the Hellenized Egyptian mathematician Menelaus of Alexandria (ca. 100 A.D.) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.[9] He establishes a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.[9] Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.[9] Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".[9] He further gave his famous "rule of six quantities".[13]

Later, the Hellenized Egyptian mathematician Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity.[14] A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.[14] Ptolemy further derived the equivalent of the half-angle formula .[14] Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.[14]

Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.[15]

Classical trigonometry

Classical Indian mathematics

Statue of Aryabhata. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.

The next significant developments of trigonometry were in India. Influential works from the 4th–5th century, known as the Siddhantas (of which there were five, the most complete survivor of which is the Surya Siddhanta[16]) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine.[17] Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya.[18] The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.[19] They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.

(a trigonometric equivalent to formulas known by Thales and Pythagoras[20])

(modern sine and cosine equivalent to a chord formula known to Ptolemy; see above)

In the 7th century, Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%:Template:Fix/category[citation needed]

Classical Islamic mathematics

The Indian works were later translated and expanded in the classical Islamic world by Islamic mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."[22]

In addition to Indian works, Hellenistic methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems.[9][23] However, E. S. Kennedy points out that while it was possible in pre-lslamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice.[24] In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menalaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.[25]

By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Islamic mathematicians were using all six trigonometric functions.[31] Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.[31] He also developed the following trigonometric formula:

Abū al-Wafā also established the angle addition identities, e.g. , and discovered the law of sines for spherical trigonometry:[29]

Islamic Spain

The book of unknown arcs of a sphere "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.[32]

In the 13th century, Nasīr al-Dīn al-Tūsī (1201-1274) was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.[30] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.[34]

Chinese mathematics

In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty.[35] Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Hellenistic, Indian and Islamic worlds.[36] Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.[35] However, this embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.[35] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[35] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle given the diameter , sagita , and length of the chord subtending the arc, the length of which he approximated as .[37] Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).[38] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[35][39] Along with a later 17th century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).[40]

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).[41]

He also gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°.[42]

Ulugh Beg (1394–1449) gave accurate tables of sines and tangents correct to 8 decimal places, around the same time as al-Kāshī.[44]

Kerala school of Indian mathematics

Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the power series expansions of sine, cosine, tangent, and arctangent.[45][46] Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the θ, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.[45][46]

No.

Series

Name

Western discoverers of the seriesand approximate dates of discovery[47]

The Indian text the Yuktibhāṣā contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.

Medieval European mathematics

Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,[48] in his De triangulis omnimodus written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed.

Modern trigonometry

Ottoman school of Islamic mathematics

In the 16th century, the Ottoman Islamic mathematician Taqi al-Din contributed to trigonometry in his Sidrat al-Muntaha, in which he was the first mathematician to extract the precise value of Sin 1°. He discusses the values given by his predecessors, explaining how Ptolemy used an approximate method to obtain his value of Sin 1° and how Abū al-Wafā, Ibn Yunus, al-Kashi, Qāḍī Zāda al-Rūmī, Ulugh Beg and Mirim Chelebi improved on the value. Taqi al-Din then solves the problem to obtain the precise value of Sin 1°:[49]

Modern European mathematics

The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In the 17th century, Isaac Newton and James Stirling developed the general Newton-Stirling interpolation formula for trigonometric functions.

In the 18th century, Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, defining them as infinite series and presenting "Euler's formula" . Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.

Also in the 18th century, Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.

See also

Citations and footnotes

↑ 2.02.12.2Boyer (1991). "Greek Trigonometry and Mensuration". pp. 166–167. "It should be recalled that form the days of Hipparchus until modern times there were no such things as trigonometric ratios. The Greeks, and after them the Hindus and the Arabs, used trigonometric lines. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values (or approximations) with the chords. [...] It is not unlikely that the 260-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans." A cycle of the seasons of roughly 360 days could readily be made to correspond to the system of zodiacal signs and decans by subdividing each sign into thirty parts and each decan into ten parts. Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty partes minutae primae, each of these latter into sixty partes minutae secundae, and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived. It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into 120 parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds."

↑Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-387-95136-9

↑ 5.05.15.2Boyer (1991). "Greek Trigonometry and Mensuration". pp. 158–159. "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians."

↑ 9.09.19.29.39.49.5Boyer (1991). "Greek Trigonometry and Mensuration". p. 163. "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue - that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest."

↑ 10.010.1Boyer (1991). "Greek Trigonometry and Mensuration". p. 159. "Instead we have an Aristarchan treatise, perhaps composed earlier (ca. 260 B.C.), On the Sizes and Distances of the Sun and Moon, which assumes a geocentric universe. In this work Aristarchus made the observation that when the moon is just half-full, the angle between the lines of sight to the sun and the moon is less than a right angle by one thirtieth of a quadrant. (The systematic introduction of the 360° circle came a little later. In trigonometric language of today this would mean that the ratio of the distance of the moon to that of the sun (the ration ME to SE in Fig. 10.1) is sin 3°. Trigonometric tables not having being developed yet, Aristarchus fell back upon a well-known geometric theorem of the time which now would be expressed in the inequalities sin α/ sin β < α/β < tan α/ tan β, where 0° < β < α < 90°.)"

↑ 11.011.1Boyer (1991). "Greek Trigonometry and Mensuration". p. 162. "For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted."

↑Boyer (1991). "Greek Trigonometry and Mensuration". p. 162. "It is not known just when the systematic use of the 360° circle came into mathematics, but it seems to be due largely to Hipparchus in connection with his table of chords. It is possible that he took over from Hypsicles, who earlier had divided the day into parts, a subdivision that may have been suggested by Babylonian astronomy."

↑ 14.014.114.214.3Boyer (1991). "Greek Trigonometry and Mensuration". pp. 164–166. "The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. [...] Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from A.D. 127 to 151 and, therefore, assume that he was born at the end of the first century. Suidas, a writer who lived in the tenth century, reported that Ptolemy was alive under Marcus Aurelius (emperor from A.D. 161 to 180).Ptolemy's Almagest is presumed to be heavily indebted for its methods to the Chords in a Circle of Hipparchus, but the extent of the indebtedness cannot be reliably assessed. It is clear that in astronomy Ptolemy made use of the catalogue of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguioshed predecrssor cannot be determined. [...] Central to the calculation of Ptolemy's chords was a geometric proposition still known as "Ptolemy's theorem": [...] that is, the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. [...] A special case of Ptolemy's theorem had appeared in Euclid's Data (Proposition 93): [...] Ptolemy's theorem, therefore, leads to the result . Similar reasoning leads to the formula [...] These four sum-and-difference formulas consequently are often known today as Ptolemy's formulas.It was the formula for sine of the difference - or, more accurately, chord of the difference - that Ptolemy found especially useful in building up his tables. Another formula that served him effectively was the equivalent of our half-angle formula."